On (1,2)*-πgθ-CLOSED SETS IN BITOPOLOGICAL SPACES
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On (1,2)*-πgθ-CLOSED SETS IN BITOPOLOGICAL SPACES

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  • 1. ISSN (e): 2250 – 3005 || Vol, 04 || Issue, 8 || August – 2014 || International Journal of Computational Engineering Research (IJCER) www.ijceronline.com Open Access Journal Page 6 On (1,2)*-πgθ-CLOSED SETS IN BITOPOLOGICAL SPACES C.Janaki1 , M. Anandhi2 1,2Asst. Professor, Dept. of Mathematics, L.R.G Government Arts College For Women, Tirupur-4, India. I. INTRODUCTION Velicko[24] introduced the notions of θ-open subsets, θ-closed subsets and θ-closure, for the sake of studying the important class of H-closed spaces in terms of arbitrary filterbases. Dontchev and Maki [7] alone have explored the concept of θ-generalized closed sets. Regular open sets have been introduced and investigated by Stone [23]. Levine [4,14] introduced generalized closed sets and studied their properties. Bhattacharya and Lahiri [5], Arya and Nour [4], Maki et al.[15],[16] introduced semi-generalized closed sets, generalized semi-closed sets and α–generalized closed sets and generalized α–closed sets respectively. O.Ravi et al [21] have introduced the concepts of (1,2)*-semi-open sets, (1,2)*-α-open sets, (1,2)*-semi-generalized-closed sets and (1,2)*-α-generalized closed sets in bitopological spaces. This paper is an attempt to highlight a new type of generalized closed sets called (1,2)*-π generalized θ-closed (briefly (1,2)*-πgθ-closed) sets and a new class of generalized functions called (1,2)*-πgθ-continuous functions and (1,2)*-πgθ-irresolute functions. These findings result in procuring several characterizations of (1,2)*-πgθ-closed sets and as well as their application which leads to an introduction of a new space called (1,2)*-πgθ-T½ space. II. PRELIMINARIES Throughout this paper (X,τ1,τ2)and (Y,σ1,σ2) represent bitopological spaces on which no separation axioms are assumed unless otherwise mentioned. Definition 2.1 ([20]). A subset S of a bitopological space (X,τ1,τ2) is said to be τ1,2-open if S = A B where A τ1and B τ2. A subset S of X is τ1,2 - closed if the complement of S is τ1,2-open. Definition 2.2 ([20]). Let S be a subset of X. Then (i) The τ1τ2–interior of S , denoted by τ1τ2 –int (S) is defined by  {G/G S and G is τ1,2 -open } . (ii) The τ1τ2–closure of S denoted by τ1τ2-cl(S) is defined by ∩{ F / S  F and F is τ1,2 -closed}. Definition 2.3 . A subset A of a bitopological space (X,τ1,τ2) is called 1. (1,2)* - semi-open[20] if A  τ1τ2-cl(τ1τ2- int(A)). 2. (1,2)*- preopen [20] if A τ1τ2- int (τ1τ2-cl(A)). 3. (1,2)*- α-open [20] if A τ1τ2- int (τ1τ2-cl (τ1τ2- int(A))). 4. (1,2)*-generalised closed (briefly (1,2)*-g-closed) [20] if τ1τ2-cl (A)  U whenever A  U and U is τ1,2 -open in X. 5. (1,2)*-regular open[20] if A= τ1τ2- int (τ1τ2-cl(A)). 6. (1,2)* - semi-generalised-closed (briefly (1,2)*-sg-closed) [20] if (1,2)*-scl(A)  U whenever A  U and U is (1,2)*-semi-open in X. 7. (1,2)*-generalized semi-closed (briefly (1,2)*-gs-closed)[20] if (1,2)*-scl(A)  U, whenever A U and U is τ1,2-open in X . Abstract In this paper we introduce a new class of sets called (1,2)*-πgθ-closed sets in bitopological spaces. Also we find some basic properties of (1,2)*-πgθ-closed sets. Further, we introduce a new space called (1,2)*-πgθ-T½ space. Mathematics Subject Classification: 54E55, 54C55 KEY WORDS: (1,2)*-πgθ-closed set, (1,2)*-πgθ-open set, (1,2)*-πgθ-continuity and (1,2)*-πgθ-T½ space.
  • 2. On (1,2)*-Πgθ-Closed Sets… www.ijceronline.com Open Access Journal Page 7 8. (1,2)*-α-generalized-closed (briefly (1,2)*-αg-closed) [20] if (1,2)*-αcl(A)  U, whenever A U and U is τ1,2 -open in X. 9. (1,2)*-generalized α-closed (briefly (1,2)*-gα-closed)[20] if (1,2)*- αcl(A)  U, whenever A  U and U is (1,2)*-α -open in X. 10. a (1,2)*-θ-generalized closed (briefly, (1,2)*-θg-closed) set [11] if (1,2)*-clθ(A)  U whenever A U and U is τ1,2 -open in (X,τ1,τ2). 11. (1,2)*-π generalized closed(briefly (1,2)*-πg-closed [8] if (1,2)*-cl(A)  U, whenever A U and U is τ1,2 -π-open. 12. (1,2)*-π generalized α-closed (briefly (1,2)*-πgα-closed)[3] if (1,2)*-αcl(A)  U, whenever A U and U is τ1,2 -π-open. 13. (1,2)*-π generalized semi-closed (briefly (1,2)*-πgs-closed)[4] if (1,2)*-scl(A)  U, whenever A U and U is τ1,2 -π-open. 14. (1,2)*-π generalized b-closed (briefly (1,2)*-πgb-closed)[22] if (1,2)*-bcl(A)  U, whenever A U and U is τ1,2 -π-open. 15. (1,2)*-π generalized pre-closed( briefly (1,2)*-πgp-closed)[19] if (1,2)*-pcl(A)  U, whenever A U and U is τ1,2 -π-open. The complement of a (1,2)*-semi-closed (resp. (1,2)*-α-closed, (1,2)*-g-closed, (1,2)*-sg-closed, (1,2)*- gs-closed, (1,2)*-αg-closed (1,2)*-gα-closed, (1,2)*-θg-closed , (1,2)* πg-closed, (1,2)*- πgα-closed, (1,2)*- πgs-closed, (1,2)*-πgb-closed, (1,2)*-πgp-closed) set is called (1,2)*-semi open(resp. (1,2)*-α-open,(1,2)*-g-open,( 1,2)*-sg-open,(1,2)*-gs-open,(1,2)*-αg-open, (1,2)*-gα-open, (1,2)*-θg-closed, (1,2)*-πg-open,(1,2)*- πgα-open,(1,2)*-πgs-open,(1,2)*-πgb-open, (1,2)*-πgp-open). Definition 2.4 The finite union of (1,2)*-regular open sets[5] is said to be τ1,2-π-open. The complement of τ1,2-π-open is said to be τ1,2-π-closed. Definition2. 5: A function f: (X,τ1,τ2) →(Y,σ1,σ2) is called (i) τ1,2-π-open map[3] if f(F) is τ1τ2-π-open map in Y for every τ1,2-openset F in X. (ii) (1,2)*-θ-continuous[7] if f-1 (V) is (1,2)*-θ-closed in (X,τ1,τ2) for every (1,2)*-closed set V in (Y,σ1,σ2). (iii) (1,2)*-θ-irresolute[7] if f-1 (V) is (1,2)*-θ-closed in (X,τ1,τ2) for every (1,2)*-θ-closed set V in (Y,σ1,σ2). III. (1,2)*- πgθ-closed set We introduce the following definition. Definition 3.1. A subset A of (X,τ1,τ2) is called(1,2)*- π generalized θ-closed set (briefly (1,2)*-πgθ-closed ) if τ1τ2-clθ(A)  U whenever A U and U is τ1τ2 -π-open. The complement of (1,2)*-πgθ-closed is (1,2)*-πgθ-open.. Theorem 3.2: 1. Every (1,2)*-θ- closed set is (1,2)*-πgθ-closed. 2. Every (1,2)*-θg-closed set is (1,2)*-πgθ-closed. 3. Every (1,2)*-πgθ-closed set is (1,2)*-πg-closed. 4. Every (1,2)*-πgθ-closed set is (1,2)*-πgα -closed. 5. Every (1,2)*-πgθ-closed set is (1,2)*-πgs -closed. 6. Every (1,2)*-πgθ-closed set is (1,2)*-πgb-closed. 7. Every (1,2)*-πgθ-closed set is (1,2)*-πgp-closed Proof: Straight forward. Converse of the above need not be true as seen in the following examples. Example 3.3 Let X ={a,b,c}.τ1={ ϕ, X ,{a},{c},{a,c} }; τ2={ϕ, X,{ b,c }}: Let A ={b}. Then A is πgθ-closed but not θ-closed. Example3.4 Let X={a,b,c,d}.τ1={ϕ,{b},{d},{b,d},{a,b,d},X}; τ2={ϕ,{b,d},{b,c,d}, X}. Let A = {a,d}. Then A is (1,2)*-πgθ-closed but not (1,2)*-θg-closed.
  • 3. On (1,2)*-Πgθ-Closed Sets… www.ijceronline.com Open Access Journal Page 8 Example 3.5 Let X = { a,b,c,d}, τ1 = {ϕ,{a},{b},{a,b},{a,b,d},X }, τ2={ ϕ ,{a,c}, {a,b,c},X}. Let A = {c}. Then A is (1,2)*-πg-closed but not (1,2)*-πgθ-closed. Example 3.6 Let X = {a,b,c,d}. τ1= {ϕ,{a},{d},{a,d}, X }; τ2 ={ ϕ,{c,d}, {a,c,d}, X}: Let A ={c}. Then A is (1,2)*-πgα-closed set but not (1,2)*-πgθ-closed. Example3.7 Let X = {a,b,c,d}. τ1= {ϕ,{a},{b},{a,b}, X}; τ2 = { ϕ,{a,b}, {a,b,d}, X,}; Let A ={a}. Then A is (1,2)*-πgs-closed set but not (1,2)*-πgθ-closed. Example 3.8 Let X={a,b,c,d}.τ1= {ϕ,{a},{d},{a,d}, X }; τ2 = {ϕ,{c,d}, {a,c,d}, X}; Let A ={a,c}. Then A is (1,2)*-πgb-closed set but not (1,2)*-πgθ-closed. Example 3.9 Let X = {a,b,c,d}. τ1= {ϕ,{a},{d},{a,d}, X }; τ2 ={ ϕ,{c,d},{a,c,d}, X}: Let A = {c}. Then A is (1,2)*-πgp-closed but not (1,2)*-πgθ-closed. Remark 3.10 The above discussions are summarized in the following diagram. . Remark 3.11 (1,2)*-πgθ-closed is independent of (1,2)*-closedness, (1,2)*-α-closedness, (1,2)*-semi- closedness, (1,2)*-sg-closedness, (1,2)*-gs-closedness, (1,2)*-g-closedness, (1,2)*-αg-closedness and(1,2)*-gα- closedness, as seen in the following examples. Example 3.12 Let X = {a,b,c,d}, τ1= {ϕ,{a},{b},{a,b}, X }; τ2 = {ϕ, {a,b,d}, X}: Let A = {d}. Then A is (1,2)*-πgθ-closed but not (1,2)*-g-closed. Example 3.13 Let X = {a,b,c,d,e}, τ1= {ϕ,{a,b},{a,b,c,d}, X}; τ2= {ϕ,{c,d},X}: Let A = {e}. Then A is (1,2)*-g-closed but not (1,2)*-πgθ-closed. Example3.14. Let X = {a,b,c,}, τ1= { ϕ,{a},{b},{a,b}, X }; τ2 = {ϕ, {b,c}, X}: Let A ={b}. Then A is (1,2)*- πgθ-closed but not (1,2)*-closed, (1,2)*-α-closed, (1,2)*-semi-closed. Example 3.15 Let X = {a,b,c}, τ1= {ϕ,{a},{b},{a,b}, X }; τ2 = { ϕ, {b,c}, X}: Let A ={a}. Then A is (1,2)*- closed, (1,2)*-α-closed, (1,2)*-semi-closed but not (1,2)*-πgθ-closed. Example 3.16 Let X = {a,b,c,d}, τ1= {ϕ,{a},{b},{a,b},{a,b,c}, X }; τ2 = {ϕ, {a,b,d}, X}: (i) Let A={a,b,c}. Then A is (1,2)*-πgθ-closed but neither (1,2)*-sg-closed nor (1,2)*-gs-closed. (ii) Let A = {a}. Then A is both (1,2)*-sg-closed and (1,2)*-gs-closed but not (1,2)*-πgθ-closed. Example 3.17 Let X = {a,b,c,d}, τ1={ϕ,{c,d},{a,c,d}, X), τ2={ϕ,{a},{d},{a,d},{c,d},X}. (i) Let A={b,d}. Then A is (1,2)*-πgθ-closed but neither (1,2)*-αg-closed nor (1,2)*-gα- closed. (ii)Let A ={c}.Then A is (1,2)*-αg-closed,(1,2)*-gα-closed but not (1,2)*-πgθ-closed. Remark 3.18 The above discussions are summarized in the following diagram. (1,2)*-πgθ-closed (1,2)*-θ-closed (1,2)*-θg-closed (1,2)*-πg-closed (1,2)*-πgα-closed (1,2)*-πgb-closed (1,2)*-πgp-closed (1,2)*-πgs-closed
  • 4. On (1,2)*-Πgθ-Closed Sets… www.ijceronline.com Open Access Journal Page 9 Remark 3.19 A finite union of (1,2)*-πgθ -closed sets is always a (1,2)*-πgθ-closed. Proof: Let A, B ϵ (1,2)*-πGθC(X). Let U be any τ1τ2-π-open set such that (A  B)  U. Since (1,2)*- clθ(A  B) = (1,2)*-clθ(A)  (1,2)*-clθ(B)  U U=U. This implies (1,2)*-clθ(A B)  U. Hence A B is also a (1,2)*-πgθ-closed set. Remark 3.20 The intersection of two (1,2)*-πgθ-closed sets need not be (1,2)*-πgθ-closed as seen in the following example. Example3.21 Let X = {a,b,c,d}, τ1 = {ϕ,{a},{b},{a,b},X }; τ2 = {ϕ,{a,b,d},X}: Let A={a,b,c} and B ={ a,b,d}.Clearly A and B are (1,2}*-πgθ-closed sets. But A∩B = { a,b} is not a (1,2)*- πgθ-closed set. Proposition 3.22 Let A be (1,2)*-πgθ-closed in (X,τ). Then (1,2)*-clθ(A)–A does not contain any non-empty τ1τ2-π-closed set. Proof: Let F be a non-empty τ1τ2-π-closed set such that F  (1,2)*-clθ(A)–A. Since A is (1,2)*-πgθ-closed, A X–F where X–F is τ1τ2-π-open implies (1,2)*-clθ(A)  (X–F). Hence F  X ̶ (1,2)*-clθ(A). Now, F  (1,2)*-clθ(A)∩X–(1,2)*-clθ(A) implies F= ϕ which is a contradiction. Therefore clθ(A)–A does not contain any non-empty τ1τ2-π-closed set. Remark 3.23 The converse of Proposition 3.22 need not be true as shown in the following example. Example 3.24 Let X = {a,b,c}. τ1= {ϕ, X,{b} }; τ2 = { ϕ, X,{c}}: Let A = {b,c}. Then (1,2)*-clθ(A) ̶ A contains no non-empty τ1τ2-π-closed set. However A is not (1,2)*-πgθ- closed. Proposition 3.25 If A is a τ1,2-regular open and (1,2)*-πgθ-closed subset of (X,τ1,τ2), then A is a (1,2)*-θ- closed subset of (X,τ1,τ2). Proof. Since A is τ1,2-regular open and (1,2)*-πgθ-closed, (1,2)*-clθ(A)  A . Hence A is (1,2)*-θ-closed. Proposition3.26 Let A be a (1,2)*-πgθ-closed subset of (X,τ1, τ2). If A  B  (1,2)*- clθ(A), then B is also a (1,2)*-πgθ-closed subset of (X,τ1,τ2). Proof. Let U be a τ1τ2-π-open set of (X,τ1, τ2) such that B U. Then A U. Since A is a (1,2)*-πgθ-closed set, (1,2)*-clθ(A)  U. Also since B (1,2)*-clθ(A), (1,2)*-clθ(B)  (1,2)* - clθ((1,2)*-clθ(A))=(1,2)*- clθ(A).Thus (1,2)*-clθ(B)  U. Hence B is also a (1,2)*-πgθ-closed subset of (X,τ1,τ2). Theorem 3.27 Let A be a (1,2)*-πgθ-closed sub set in X. Then A is (1,2)*-θ-closed if and only if (1,2)*- clθ(A)–A is τ1,2 -π-closed. Proof. Necessity: Let A be (1,2)*-θ-closed subset of X. Then (1,2)*-clθ(A)=A and (1,2)*-clθ(A)–A=ϕ which is τ1,2 -π-closed. Sufficiency: Since A is (1,2)*-πgθ-closed, by proposition 3.22 (1,2)*-clθ(A–A contains no non-empty τ1,2- π- closed set. But (1,2)*-clθ(A) – A is τ1,2-π-closed. This implies (1,2)*-clθ(A) – A=ϕ, which means (1,2)*- clθ(A)=A and hence A is (1,2)*-θ-closed.
  • 5. On (1,2)*-Πgθ-Closed Sets… www.ijceronline.com Open Access Journal Page 10 IV. (1,2)*-πgθ -open sets Definition 4.1 A subset A of (X, τ1, τ2) is called (1,2)*-πgθ-open if and only if Ac is (1,2)*-πgθ-closed in (X,τ1,τ2). Remark 4.2 For a subset A of (X, τ1, τ2), (1,2)*-clθ(Ac) = [(1,2)*- intθ(A)]c Theorem 4.3 A subset A of (X,τ1,τ2) is (1,2)*-πgθ-open if and only if F  (1,2)*-intθ(A) whenever F is τ1 τ2-π- closed and F  A. Proof. Necessity: Let A be a (1,2)*-πgθ-open set in (X,τ1,τ2). Let F be τ1τ2-π-closed and F A. Then Fc  Ac and Fc is τ1τ2-π-open. Since Ac is (1,2)*-πgθ-closed, (1,2)*-clθ(Ac)  Fc. By remark 4.2, [(1,2)*-intθ(A)]c  Fc. That is F (1,2)*-intθ(A). Sufficiency: Let Ac  U where U is τ1τ2-π-open. Then Uc  A where Uc is (1,2)*-τ1τ2-π-closed. By hypothesis Uc  (1,2)*-intθ(A ). That is [(1,2)*-intθ(A}]c  U. By remark 4.2, (1,2)*-clθ(Ac)  U. This implies Ac is (1,2)*-πgθ-closed. Hence A is (1,2)*-πgθ-open. Theorem 4.4 If (1,2)*-intθ(A)  B A and A is (1,2)*-πgθ-open, then B is also (1,2)*-πgθ-open. Proof. (1,2)*-intθ(A)  B A implies Ac  Bc  [(1,2)*-intθ(A)]c . By remark 4.2 Ac  Bc  (1,2)*-clθ(Ac). Also Ac is (1,2)*-πgθ-closed. By Proposition 3.26 Bc is (1,2)*-πgθ-closed. Hence B is (1,2)*-πgθ-open. As an application of (1,2)*-πgθ-closed sets we introduce the following definition. Definition 4.5 A space (X,τ1, τ2) is called a (1,2)*-πgθ-T½ space if every (1,2)*-πgθ-closed set is (1,2)*-θ- closed. Theorem 4.6 For a space (X,τ1,τ2) the following conditions are equivalent. (i) (X,τ1,τ2) is a (1,2)*-πgθ-T½ space. (ii) Every singleton set of (X,τ1,τ2) is either τ1τ2-π-closed or (1,2)*-θ-open. Proof. (i)  (ii). Let x  X. Suppose {x} is not a τ1τ2-π- closed set of (X, τ1, τ2). Then X–{x} is not a τ1τ2-π- open set. So X is the only τ1τ2-π-open set containing X–{x}.So X-x is a (1,2)*-πgθ-closed set of (X,τ1,τ2). Since (X,τ1,τ2) is a (1,2)*-πgθ-T½ space, X– x is a (1,2)*-θ-closed set of (X,τ1,τ2) or equivalently {x} is a (1,2)*-θ- open set of (X,τ1, τ2). (ii)  (i). Let A be a (1,2)*-πgθ-closed set of X. Trivially A (1,2)*-clθ(A). Let xϵ1,2)*-clθ(A). By (ii) {x} is either τ1τ2-π-closed or (1,2)*-θ-open. (a) Suppose that {x} is τ1 τ2-π-closed. If x A, then xє(1,2)*-clθ(A)–A contains a non-empty τ1τ2-π-closed set {x}. By proposition 3.6 we arrive at a contradiction. Thus xϵA. (b) Suppose that {x} is a (1,2)*-θ-open. Since xϵ(1,2)*-clθ(A)=A or equivalently A is (1,2)*-θ-closed. Hence (X,τ1,τ2) is a (1,2)*-πgθ-T½ space. V. (1,2)*-πgθ-continuous and (1,2)*-πgθ-irresolute functions Definition 5.1 A function f:(X,τ1,τ2) →(Y,σ1,σ2) is called (1,2)*-πgθ-continuous if every f-1(V) is (1,2)*-πgθ- closed in (X,τ1, τ2) for every(1,2)*-σ1σ2-closed set V of (Y,σ1,σ2). Definition 5.2 A function f:(X,τ1,τ2)→(Y,σ1,σ2) is called (1,2)*-πgθ-irresolute if f-1 (V) is (1,2)*-πgθ-closed in (X,τ1,τ2) for every (1,2)*-πgθ-closed set V in (Y,σ1,σ2). Remark 5.3 (1,2)*-πgθ-irresolute function is independent of (1,2)*-θ-irresoluteness, as seen in the following examples. Example 5.4 Let X=Y={a,b,c,d}, τ1={ ϕ,{a},{d},{a,d},{a,c,d},X},τ2= {ϕ,{a,d},{a,b,d}, X}, σ1 = {ϕ,{a},{d},{a,d},X},σ2= {ϕ,{c,d},{a,c,d},X}. Let f: (X,τ1,τ2)→(Y,σ1,σ2) be an identity function. Then f is (1,2)*-θ-irresolute but not (1,2)*-πgθ-irresolute, since f-1 [{b,c,d}]={b,c,d} is not (1,2)*-πgθ-closed in (X,τ1,τ2).
  • 6. On (1,2)*-Πgθ-Closed Sets… www.ijceronline.com Open Access Journal Page 11 Example 5.5 Let X=Y={a,b,c,d}, τ1={ϕ,{a},{d},{a,d},{a,c,d},X }, τ2 = {ϕ,{a,d}, {a,b,d},X}, σ1 ={ϕ,{a},{c},{a,c}, {c,d},{a,c,d},X},σ2= {ϕ,{d},{a,b,d},X}, Let f: (X,τ1,τ2)→(Y,σ1,σ2) be an identity function. Then f is (1,2)*-πgθ-irresolute but not (1,2)*-θ-irresolute, since f-1 [{a,b,d}] = {a,b,d} is not (1,2)*-θ-closed in (X,τ1,τ2). Remark 5.6: Every (1,2)*-θ-continuous is (1,2)*-πgθ-continuous. The converse of the above need not be true as seen in the following examples. Example 5.7: Let X=Y={a,b,c,d,e }, τ1={ ϕ,{a,b},{a,b,c},X }, τ2= {ϕ,{c},X},σ1 ={ ϕ, {a,b}, {a,b,c,d}, X },σ2={ ϕ,{c,d},{a,b,c,d}, X }.Let f: (X,τ1,τ2)→(Y,σ1,σ2) be an identity function. Then f is (1,2)*-πgθ-continuous but not (1,2)*-θ-continuous, since f-1[{c,d,e}]={c,d,e} is not (1,2)*-θ-closed in (X, τ1, τ2). Remark 5.8 (1,2)*-πgθ-continuous is independent of (1,2)*- πgθ-irresolute as seen in the following examples. Example 5.9 Let X=Y={a,b,c,d,e }, τ1={ ϕ,{a,b},{a,b,c,d},X }, τ2= {ϕ,{c,d},X},σ1 ={ ϕ,{b}, {b,c}, X },σ2={ ϕ,{c},{a,b},{a,b,c}, X }.Let f: (X,τ1,τ2)→(Y,σ1,σ2) be an identity function. Then f is (1,2)*-πgθ-continuous but not (1,2)*-πgθ-irresolute, since f-1[{d}]={d} is not (1,2)*-πgθ-closed in (X, τ1, τ2) where {d}is πgθ-closed in (Y,σ1,σ2). Example 5.10: Let X=Y={a,b,c,d}, τ1={ϕ,{a},{b},{a,b},{a,b,c},X }, τ2= {ϕ,{b},{c},{b,c},{a,c}, {a,b,c},{a,b,d},X},σ1 ={ ϕ, {a},{b},{a,b},{a,d}, {a,b,d},X },σ2={ ϕ,{c},{a,c},{b,c},{a,b,c}, {a,c,d}, X }.Let f: (X,τ1,τ2)→(Y,σ1,σ2) be an identity function. Then f is (1,2)*-πgθ-irresolute but not (1,2)*-πgθ-continuous, since f-1[{b}]={b} is not (1,2)*-πgθ-closed in (X, τ1, τ2) where {b}is closed in (Y,σ1,σ2). Remark 5.11 The above discussions are summarized in the following diagram. Where (1)  (1,2)*-θ-continuous; (2)  (1,2)*-πgθ-continuous; (3)  (1,2)*-πgθ-irresolute; (4)  (1,2)*-θ-irresolute. Remark 5.12 Composition of two (1,2)*-πgθ-continuous function need not be (1,2)*-πgθ-continuous. Example 5.13 Let X=Y=Z={a,b,c,d,e},τ1={ϕ,{a,b},{a,b,c},X}, τ2= {ϕ,{c},X},σ1 ={ϕ,{a,b},X}, σ2 ={ ϕ,{c,d},{a,b,c,d},X}, η1={ϕ,{a,b,c,d},X},η2 ={ ϕ ,{e}, X}. Let f:(X,τ1,τ2)→(Y,σ1,σ2) and g:(Y,σ1,σ2)→(Z,η1,η2) be the identity functions. Both f and g are (1,2)*-πgθ-continuous but gof is not (1,2)*-πgθ-continuous, since (gof)-1[{a,b,c,d}]={a,b,c,d} is not (1,2)*-πgθ-closed in (X,τ1,τ2). Theorem 5.14 Let f: (X,τ1, τ2)→(Y,σ1,σ2) be a function . (i) If f is (1,2)*-πgθ-irresolute and X is(1,2)*-πgθ-T½ space, then f is(1,2)*-θ-irresolute. (ii) If f is (1,2)*-πgθ-continuous and X is (1,2)*-πgθ-T½ space then f is (1,2)*-θ-continuous. Proof: (i) Let V be (1,2)*-θ-closed in Y. Since f is (1,2)*-πgθ-irresolute, f-1(V) is (1,2)*- πgθ-closed in X. Since X is (1,2)*-πgθ-T½ space, f-1(V) is (1,2)*-θ-closed in X. Hence f is (1,2)*-θ-irresolute. (ii)Let V be closed in Y. Since f is (1,2)*-πgθ-continuous, f-1(V) is (1,2)*-πgθ-closed in X. By assumption, it is (1,2)*-θ-closed. Therefore f is (1,2)*-θ-continuous. Theorem 5.15 If the bijective f: (X,τ1,τ2)→( Y,σ1,σ2) is (1,2)*-θ-irresolute and τ1τ2-π-open map, then f is (1,2)*-πgθ-irresolute.
  • 7. On (1,2)*-Πgθ-Closed Sets… www.ijceronline.com Open Access Journal Page 12 Proof: Let V be (1,2)*-πgθ-closed in Y. Let f-1(V)  U where U is τ1τ2-π-open in X. Then V f(U) and f(U) is τ1τ2-π-open implies (1,2)*-clθ(V)  f(U). This implies f-1( (1,2)*-clθ(V))  U. Since f is (1,2)*-θ-irresolute, f- 1((1,2)*-clθ(V)) is (1,2)*-θ-closed. Hence (1,2)*- clθ(f-1(V))  (1,2)*- clθ(f-1((1,2)*-clθ(V)))=f-1((1,2)*- clθ(V))  U. Therefore f is (1,2)*-πgθ-irresolute. Theorem 5.16 :If f: (X,τ1,τ2)→(Y,σ1,σ2) is (1,2)*-πgθ-irresolute map and g: (Y,σ1,σ2)→ (Z,η1,η2) is (1,2)*- πgθ-continuous map, the composition gof: (X,τ1,τ2)→ (Z,η1,η2) is a (1,2)*-πgθ-continuous map. Proof: Let V be η1η2–closed set in Z. Since g:(Y,σ1,σ2)→(Z,η1,η2) is a (1,2)*-πgθ-continuous map,g-1(V) is (1,2)*-πgθ-closed in Y. By hypothesis ,f-1(g-1(V))=(gof)-1(V) is (1,2)*-πgθ-closed in X. Hence gof: (X,τ1,τ2) → (Z,η1,η2) is a (1,2)*-πgθ-continuous map. V. CONCLUSION Through the above findings, this paper has attempted to compare (1,2)*-πgθ-closed with the other closed sets in bitopological spaces. An attempt of this paper is to state that the several definitions and results that shown in this paper, will result in obtaining several characterizations and enable to study various properties as well. It brings to limelight that the weaker form of continuity in bitopological settings is the future scope of study. REFERENCES [1] M. E. Abd El-Monsef, S.Rose Mary and M.Lellis Thivagar (1,2)*- α g  -closed sets in topological spaces, Assiut Univ. J. of Mathematics and Computer Science, Vol.36(1) (2007), 43-51. [2] M. Anandhi and C.Janaki: On πgθ-Closed sets in Topological spaces, International Journal of Mathematical Archive-3(11), 2012, 3941-3946. [3] I. Arockiarani and K. Mohana, (1,2)*- πgα-closed sets and (1,2)*-Quasi-α-normal Spaces in Bitopological settings, Antarctica j. Math.,7(3)(2010), 3465-355. [4] S. P. Arya and T. Nour: Characterizations of S-normal spaces, Indian J. Pure. Appl. Math., 21(1990), No. 8, 717-719. [5] P.Bhattacharya and B.K. Lahiri: Semi-generalised closed sets in topology, Indian J. Math., 29(1987), 375-382. [7] J. Dontchev and H. Maki,On θ-generalized closed sets,Internat.J. Math& Math.Sci. 22(2) (1999) 239-249. [8] J. Dontchev and T. Noiri, Quasi Normal Spaces and πg-closed sets, Acta Math. Hungar., 89(3)(2000). 211-219. [9] J. Dontchev and M. Przemski, The various decompositions of continuous and weakly continuous functions, Acta Math. Hungar., 71(1996), no. 1-2, 109-120. [10] E. Ekici and M . Caldas, Slightly-continuous functions, Bol. Soc. Parana. Mat. (3) 22 (2004), 63-74. [11] Govindappa Navalagi and Md. Hanif Page, θ-generalized semi-open and θ- generalized semi-closed functions Proyeceiones J. of Math. 28,2 ,(2009) 111-123. [12] C.Janaki, Studies on πgα-closed sets in Topology, Ph.D Thesis, Bharathiar University, Coimbatore (2009). [13] N.Levine: Semi-open sets and semi-continuity in topological spaces, Amer . Math. Monthly, 70 (1963), 36-41. [14] N.Levine: Generalised closed sets in topology, Rend. Circ. Mat. Palermo, 19(1970), 89-96. [15] H. Maki,R.Devi and K.Balachandran: Semi-generalised closed maps and generalized semi-closed maps, Mem. Fac. Sci. Kochi Univ. Ser. A Math.,14(1993),41-54. [16] H. Maki, R. Devi and K. Balachandran generalized α-closed maps and α-generalised closed maps, Indian J. Pure . Appl. Math., 29(1998), No. 1, 37-49. [17] A.S. Mashhour, M. E. Abd El- Monsef and S.N. El Deeb: On pre continuous and weakly pre continuous mappings, Proc. Math. Phys. Soc. Egypt., 53(1982), 47-53. [18] O.Njastad: On some Classes of nearly open sets, Pacific J. Math.,15(1965),961-970. [19] J.H. Park, On πgp-closed sets in topological spaces, Indian J. Pure. Appl. Math., (To appear). [20] O.Ravi and M.Lellis Thivagar, On Stronger forms of (1,2)* - quotient mappings on bitopological spaces,Internat. J.Math.Game Theory & Algebra4(2004),No.6,481-492. [21] O.Ravi, M.Lellis Thivagarand M.E.Abd El-Monsef, “Remarks on bitopological (1,2)*-quotient mappings”,J. Egypt Math. Soc. Vol. 16, No.1, pp.17-25,2008. [22] D.Sreeja and C.Janaki, On(1,2)*-πgb-closed sets, Inter. J. of Computer Applicationz (0975-8887), Vol.42.,No.5. [23] M.H. Stone : Application of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc., 41 (1937), 375-381. [24] N. V. Velicko, On H- closed topological spaces, Amer. Math. Soc. Transl.,78, (1968) 103-118.